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Nielsen coincidence theory in arbitrary codimensions
- J. reine angew. Math
"... Let f1,f2: M − → N be two (continuous) maps between smooth connected manifolds M and N without boundary, of strictly positive dimensions m and n, resp., M being compact. We are interested in making the coincidence locus C(f1,f2): = {x ∈ M | f1(x) = f2(x)} as small (or simple in some sense) as possi ..."
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Cited by 22 (5 self)
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Let f1,f2: M − → N be two (continuous) maps between smooth connected manifolds M and N without boundary, of strictly positive dimensions m and n, resp., M being compact. We are interested in making the coincidence locus C(f1,f2): = {x ∈ M | f1(x) = f2(x)} as small (or simple in some sense) as possible after possibly deforming f1 and f2 by a homotopy. Question. How large is the minimum number of coincidence components MCC(f1,f2): = min{#π0(C(f ′ 1,f ′ 2)) | f ′ 1 ∼ f1,f ′ 2 ∼ f2}? In particular, when does this number vanish, i.e. when can f1 and f2 be deformed away from one another? This is a very natural generalization of one of the central problems of classical fixed point theory (where M = N and f2 = identity map): determine the minimum number of fixed points among all maps in a given homotopy class (see [Br] and [BGZ], proposition 1.5). Note, however, that in higher codimensions m − n> 0 the coincidence locus is generically a closed (m−n)-manifold so that it makes more sense to count pathcomponents rather than points. Also the methods of (first order, singular) (co)homology will no longer be strong enough to capture the subtle geometry of coincidence manifolds. In this lecture I will use the language of normal bordism theory (and a nonstabilized version thereof) to define and study lower bounds N(f1,f2) (and N #(f1,f2)) for MCC(f1,f2). After performing an approximation we may assume that the map (f1,f2) : M → N × N is smooth and transverse to the diagonal ∆ = {(y,y) ∈ N × N | y ∈ N}. Then the coincidence locus C = C(f1,f2) = (f1,f2) −1 (∆) is a closed smooth (m − n)-dimensional manifold, equipped with i) maps
Kervaire invariants and selfcoincidences
, 2007
"... Abstract. Minimum numbers decide whether a given map f : S m → S n /G from a sphere into a spherical space form can be deformed to a map f such that f (x) = f (x) for all x ∈ S m . In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In th ..."
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Abstract. Minimum numbers decide whether a given map f : S m → S n /G from a sphere into a spherical space form can be deformed to a map f such that f (x) = f (x) for all x ∈ S m . In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In the stable dimension range these numbers coincide. But already in the first nonstable range (when m = 2n − 2) the Kervaire invariant appears as a decisive additional obstruction which detects interesting geometric coincidence phenomena. Similar results (involving e.g. Hopf invariants, taken mod 4) are obtained in the next seven dimension ranges (when 1 < m − 2n + 3 8). The selfcoincidence context yields also a precise geometric criterion for the open question whether the Kervaire invariant vanishes on the 126-stem or not.
Selfcoincidences and roots in Nielsen theory
"... Abstract. Given two maps f1 and f2 from the sphere S m to an n-manifold N, when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On one hand the resulting bordism class of coincidence data ..."
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Abstract. Given two maps f1 and f2 from the sphere S m to an n-manifold N, when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On one hand the resulting bordism class of coincidence data and the corresponding Nielsen numbers are strong looseness obstructions. On the other hand the values which these invariants may possibly assume turn out to satisfy severe restrictions, e.g. the Nielsen numbers can only take the values 0, 1 or the cardinality of the fundamental group of N. In order to show this we compare different Nielsen classes in the root case (where f1 or f2 is constant) and we use the fact that all but possibly one Nielsen class are inessential in the selfcoincidence case (where f1 = f2). Also we deduce strong vanishing results.
Coincidences and secondary Nielsen numbers
"... Abstract. Let f1, f2 : X m −→ Y n be maps between smooth connected manifolds of the indicated dimensions m and n. Can f1, f2 be deformed by homotopies until they are coincidence free (i.e. f1(x) = f2(x) for all x ∈ X )? The main tool for addressing such a problem is tradionally the (primary) Nielse ..."
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Abstract. Let f1, f2 : X m −→ Y n be maps between smooth connected manifolds of the indicated dimensions m and n. Can f1, f2 be deformed by homotopies until they are coincidence free (i.e. f1(x) = f2(x) for all x ∈ X )? The main tool for addressing such a problem is tradionally the (primary) Nielsen number N (f1, f2) . E.g. when m < 2n − 2 the question above has a positive answer precisely if N (f1, f2) = 0 . However, when m = 2n − 2 this can be dramatically wrong, e.g. in the fixed point case when m = n = 2 . Also, in a very specific setting the Kervaire invariant appears as a (full) additional obstruction. In this paper we start exploring a fairly general new approach. This leads to secondary Nielsen numbers SecN (f1, f2) which allow us to answer our question e.g. when m = 2n − 2, n = 2 is even and Y is simply connected. Mathematics Subject Classification (2010). Primary 54H25, 55M20; Secondary 55Q40.
Coincidence free pairs of maps
"... Abstract This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loo ..."
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Abstract This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new advances concerning the first problem. Then we attack the second problem mainly in the setting of homotopy groups. This leads also to a very natural filtration of all homotopy sets. Explicit calculations are carried out for maps into spheres and projective spaces.
SOME HOMOTOPY THEORETICAL QUESTIONS ARISING IN NIELSEN COINCIDENCE THEORY
"... Basic examples show that coincidence theory is intimately related to central subjects of differential topology and homotopy theory such as Kervaire invariants and divisibility prop-erties of Whitehead products and of Hopf invariants. We recall some recent results and ask a few questions which seems ..."
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Basic examples show that coincidence theory is intimately related to central subjects of differential topology and homotopy theory such as Kervaire invariants and divisibility prop-erties of Whitehead products and of Hopf invariants. We recall some recent results and ask a few questions which seems to be important for a more comprehensive understanding.
